The Experts below are selected from a list of 1287 Experts worldwide ranked by ideXlab platform
C I Christov - One of the best experts on this subject based on the ideXlab platform.
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fourier galerkin method for 2d solitons of boussinesq equation
Mathematics and Computers in Simulation, 2007Co-Authors: Marios A. Christou, C I ChristovAbstract:We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special Complete Orthonormal System of functions in L^2(-~,~) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible.
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Interacting localized waves for the regularized long wave equation via a Galerkin spectral method
Mathematics and Computers in Simulation, 2005Co-Authors: M. A. Christou, C I ChristovAbstract:We develop a Fourier-Galerkin spectral technique for computing the solutions of type of interacting localized waves. To this end, a special Complete Orthonormal System of functions in L^2(-~,~) is used and a time-stepping algorithm implementing the spectral method is developed. The rate of convergence of the coefficients is shown to be exponential. We consider the regularized long wave equation (RLWE) which is not fully integrable. We demonstrate the stability of the algorithm and find numerically the threshold for the existence of such interactions. We also calculate the phase shifts of the interactions and compare them to the finite-difference solution.
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Fourier‐Galerkin Method for Time Dependent Problems of Interacting Localized Waves
AIP Conference Proceedings, 2005Co-Authors: Marios A. Christou, C I ChristovAbstract:We develop a Fourier‐Galerkin spectral technique for computing solutions of type of interacting localized waves. We use a special Complete Orthonormal System of functions in L2(−∞, ∞). The rate of convergence of the coefficients is shown to be exponential. As a featured example we consider the Boussinesq Paradigm Equation (BPE). We obtain results for the head‐on and overtaking collisions of two or three solitons. We evaluate also the phase shifts of solitons.
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Fourier-Galerkin method for interacting localized waves
Neural Parallel & Scientific Computations archive, 2002Co-Authors: M. A. Christou, C I ChristovAbstract:We develop a Fourier-Galerkin spectral technique for computing the solutions of Fourth-order Generalized Wave Equations of type of interacting localized waves. To this end a special Complete Orthonormal System of functions in L2(-∞,∞) is used and a time-stepping algorithm implementing the spectral method is developed. The rate of convergence is shown to be exponential.As a featuring example the head-on collision of sech solitary waves is investigated in the case of Proper Bonssinesq Equation (PBE). It is shown that the solitons recover their exact shapes after the collision but experience phase shift. The numerically obtained signs and magnitudes of the phase shifts are in very good quantitative agreement with analytical results for the two soliton solution of PBE.
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Fourier-Galerkin Method for Localized Solutions of Equations with Cubic Nonlinearity
Journal of Computational Analysis and Applications, 2002Co-Authors: M. A. Christou, C I ChristovAbstract:Using a Complete Orthonormal System of functions in L2(−∞ ,∞) a Fourier-Galerkin spectral technique is developed for computing of the localized solutions of equations with cubic nonlinearity. A formula expressing the triple product into series in the System is derived. Iterative algorithm implementing the spectral method is developed and tested on the soliton problem for the cubic Boussinesq equation. Solution is obtained and shown to compare quantitatively very well to the known analytical one. The issues of convergence rate and truncation error are discussed.
Brian J. Mccartin - One of the best experts on this subject based on the ideXlab platform.
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Spectral structure of the equilateral triangle I: Lamé's formulas
2008Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle with either Dirichlet or Neumann boundary conditions are reviewed. The eigenfunctions are shown to form a Complete Orthonormal System. Various properties of the spectrum and nodal/antinodal lines are explored.
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Plenary lecture III: spectral structure of the equilateral triangle
2008Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle with Dirichlet and Neumann boundary conditions will be reviewed and then extended to the Robin boundary condition. The eigenfunctions will be shown to form a Complete Orthonormal System. Various properties of the spectrum and modal functions will be explored.
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Eigenstructure of the equilateral triangle. Part III. The Robin problem
International Journal of Mathematics and Mathematical Sciences, 2004Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a Complete Orthonormal System. Various properties of the spectrum and modal functions are explored.
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Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem
SIAM Review, 2003Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a Complete Orthonormal System. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.
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Eigenstructure of the Equilateral Triangle, Part I:
2003Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary math- ematical techniques. They are shown to form a Complete Orthonormal System. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.
Margit Pap - One of the best experts on this subject based on the ideXlab platform.
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Slice Regular Malmquist–Takenaka System in the Quaternionic Hardy Spaces
Analysis Mathematica, 2018Co-Authors: Margit PapAbstract:In this paper the slice regular analogue of the Malmquist–Takenaka System is introduced. It is proved that, under certain restrictions regarding to the parameters of the System, they form a Complete Orthonormal System in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are also studied.
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Slice regular Malmquist-Takenaka System in the quaternionic Hardy spaces
arXiv: Complex Variables, 2016Co-Authors: Margit PapAbstract:A slice regular analogue of the Malmquist-Takenaka System is investigated. It is proved that they form a Complete Orthonormal System in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are studied. Key Words and Phrases: Functions of hypercomplex variables, quaternionic Hardy spaces, series expansions, interpolation, approximation by rational functions, quaternionic Malmquist-Takenaka System
K. S. Kazarian - One of the best experts on this subject based on the ideXlab platform.
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A uniformly bounded Complete Euclidean System
arXiv: Classical Analysis and ODEs, 2019Co-Authors: K. S. KazarianAbstract:A uniformly bounded Complete Orthonormal System of functions $\Theta =\{ \theta_n\}_{n=1}^{\infty},$ $ \|\theta_n\|_{L^\infty_{[0,1]} } \leq M $ is constructed such that $\sum_{n=1}^{\infty} a_{n}\theta_{n}$ converges almost everywhere on $[0,1]$ if $\{ a_n\}_{n=1}^{\infty} \in \, l^2$ and $\sum_{n=1}^{\infty} a_{n}\theta_{n}$ diverges a. e. for any $\{ a_n\}_{n=1}^{\infty} \not\in \, l^2$. Thus Menshov's theorem on the representation of measurable, almost everywhere finite, functions by almost everywhere convergent trigonometric series cannot be extended to the class of uniformly bounded Complete Orthonormal Systems.
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A problem of Ul'yanov
Sbornik: Mathematics, 2006Co-Authors: K. S. KazarianAbstract:A Complete Orthonormal System of functions in is constructed such that a series converges a.e. on if , and for arbitrary the series diverges a.e. This gives a Complete answer to a problem posed by Ul'yanov.
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A Complete Orthonormal System of divergence
Journal of Functional Analysis, 2004Co-Authors: K. S. KazarianAbstract:Abstract A Complete Orthonormal System of functions {Θn}n=1∞,Θn∈L[0,1]∞, defined on the closed interval [0,1] is constructed such that ∑n=1∞anΘn diverges almost everywhere for any {a n } n=1 ∞ ∉ l 2 . For the constructed System the following result is true: Corollary 1. Any nontrivial series in the System {Θn}n=1∞ which converges in measure to zero diverges almost everywhere.
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The zero-one law for a Complete Orthonormal System
Comptes Rendus Mathematique, 2004Co-Authors: K. S. KazarianAbstract:Abstract A Complete Orthonormal System of functions Θ = { θ n } n = 1 ∞ , θ n ∈ L [ 0 , 1 ] ∞ is constructed such that ∑ n = 1 ∞ a n θ n converges almost everywhere on [ 0 , 1 ] if { a n } n = 1 ∞ ∈ l 2 and ∑ n = 1 ∞ a n θ n diverges a.e. for any { a n } n = 1 ∞ ∉ l 2 . We also show that for any Complete ONS { f n } n = 1 ∞ of functions defined on [ 0 , 1 ] there exists a fixed non decreasing subsequence { n k } k = 1 ∞ of natural numbers such that for any f ∈ L [ 0 , 1 ] 0 and some sequence of coefficients { b n } n = 1 ∞ , ∑ n = 1 n k b n f n → f a.e. when k → ∞ . To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).
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A Complete Orthonormal System of divergence
Comptes Rendus Mathematique, 2003Co-Authors: K. S. KazarianAbstract:Abstract A Complete Orthonormal System of functions {Θ n } n=1 ∞ , Θ n ∈L ∞ [0,1] defined on the closed interval [0,1] is constructed such that ∑n=1∞anΘn diverges almost everywhere for any {an}n=1∞∉l2. For the constructed System the following result is true: Any nontrivial series by the System {Θn}n=1∞ which converges in measure to zero diverges almost everywhere. To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 337 (2003).
Marios A. Christou - One of the best experts on this subject based on the ideXlab platform.
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Kawahara solitons in Boussinesq equations using a robust Christov-Galerkin spectral method
Applied Mathematics and Computation, 2014Co-Authors: Marios A. Christou, N. C. PapanicolaouAbstract:We develop a robust Christov-Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special Complete Orthonormal System of functions in L^2(-~,~) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives. As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.
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fourier galerkin method for 2d solitons of boussinesq equation
Mathematics and Computers in Simulation, 2007Co-Authors: Marios A. Christou, C I ChristovAbstract:We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special Complete Orthonormal System of functions in L^2(-~,~) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible.
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Christov‐Galerkin Expansion for Localized Solutions in Model Equations with Higher Order Dispersion
AIP Conference Proceedings, 2007Co-Authors: Marios A. ChristouAbstract:We develop a Galerkin spectral technique for computing localized solutions of equations with higher order dispersion. To this end, the Complete Orthonormal System of functions in L2(−∞,∞) proposed by Christov [1] is used.As a featuring example, the Sixth‐Order Generalized Boussinesq Equation (6GBE) is investigated whose solutions comprise monotone shapes (sech‐es) and damped oscillatory shapes (Kawahara solitons). Localized solutions are obtained here numerically for the case of the moving frame which are used as initial conditions for the time dependent problem.
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Fourier‐Galerkin Method for Time Dependent Problems of Interacting Localized Waves
AIP Conference Proceedings, 2005Co-Authors: Marios A. Christou, C I ChristovAbstract:We develop a Fourier‐Galerkin spectral technique for computing solutions of type of interacting localized waves. We use a special Complete Orthonormal System of functions in L2(−∞, ∞). The rate of convergence of the coefficients is shown to be exponential. As a featured example we consider the Boussinesq Paradigm Equation (BPE). We obtain results for the head‐on and overtaking collisions of two or three solitons. We evaluate also the phase shifts of solitons.